Typically, structural materials of various electronic apparatuses including personal computers and mobile phones are designed under the assumption that such apparatuses may be used under stresses smaller than the yield stresses of the materials thereof. Therefore, the structures of such apparatuses can be designed if their linear material properties (moduli of longitudinal elasticity, i.e., Young's moduli) and linear material characteristics are known. With the reduced sizes and thicknesses of such apparatuses in recent years, however, there has been an increasing need to design apparatuses taking into consideration situations where the apparatuses may be subjected to stresses over the yield stresses of the materials thereof.
A characteristic of a material exhibited beyond a point at which a stress applied to the material exceeds the yield stress of the material and causes plastic deformation of the material is represented by a stress-strain characteristic. FIG. 31 shows an exemplary stress-strain curve of an aluminum alloy. As shown in FIG. 31, the stress and the strain are initially proportional to each other (where Hooke's law holds) as represented by a straight line passing the origin. When, however, the stress exceeds the yield stress, the stress-strain relationship becomes nonlinear as represented by a curve shown in FIG. 31. In the example shown in FIG. 31, the stress-strain relationship becomes nonlinear under a stress of about 350 MPa. The material characteristic representing a stress-strain relationship exhibited after the linear portion is the stress-strain property, which is a material value intrinsic to each individual material.
In a typical method of measuring a stress-strain curve of, for example, a metal material, a tensile test piece conforming to JIS Z 2201 (test pieces for tensile test for metallic materials) is prepared in accordance with JIS Z 2241 (method of tensile test for metallic materials), and the stress-strain curve is measured by performing a tensile test (see Japanese Laid-open Patent Publication No. 2003-232709). In this testing method, the initial gauge length L (mm) at the time of the preparation of the test piece under no load is defined first. Subsequently, the test piece is subjected to a load P (N), which is sequentially changed. At every change in the load P (N), the load P (N) and a gauge length L′ (mm) corresponding thereto are measured, whereby a nominal strain ε is calculated in accordance with Equation (1):
A stress σ is calculated from the initial cross section A of the test piece expressed by A=w (width)×t (thickness) and each of the loads P in accordance with Equation (2):
The test piece specified in JIS Z 2201, however, is very large. FIG. 32 shows a typical example of No. 1 test piece. Therefore, measurement of materials used for electronic apparatuses, such as rare metals including gold and gold compounds, expensive resin materials, and the like, costs an impractically large amount of money. In cases of thin-film materials that can only be provided with very small thicknesses, it is difficult to produce a test piece. Brittle materials such as bismuth-based metal compounds and some resin materials undergo substantially no elongation in tensile tests, resulting in difficulties in performing tensile tests with high accuracy.
For the material characteristic represented by the linear portion of the stress-strain curve, an elastic modulus, as a bend elastic modulus, is calculated by performing a bending test, specifically, a three-point bending test, described below. FIG. 33 is a diagram for describing the three-point bending test. In the three-point bending test, a characteristic is utilized that a deflection δ occurring when a concentrated load P is applied to a double-end-supported beam is inversely proportional to the elastic modulus and is proportional to the load. The deflection δ is calculated in accordance with Equation (3):
The elastic modulus may be calculated from the deflection δ and the load P in accordance with Equation (4) obtained by solving Equation (3) for the elastic modulus E:
In Equations (3) and (4), “I” denotes the second moment of area of the test piece and is expressed by “I=bh3/12”, where b denotes the width of the test piece, and h denotes the thickness of the test piece. For example, JIS H 7406 specifies a test method for flexural properties of fiber reinforced metals.
In the bending test, since the load and the amount of deformation can be controlled more easily than in the tensile test, the elastic modulus can be measured more easily than in the tensile test. Particularly, when test pieces of substantially the same size are used in the two tests, the amount of deformation occurring in the bending test is larger than that occurring in the tensile test. Therefore, in the bending test, accurate measurement of elastic modulus can be easily performed even with a measurement apparatus having low accuracy in deformation measurement.
For example, a case of an aluminum test piece having an elastic modulus of about 70000 MPa and a rectangular shape with a thickness of 1 mm, a width of 10 mm, and a length of 100 mm will be considered. The load to produce an elongation of 1 mm in a tensile test is 7000 N (about 700 kgf) according to the following equation:
Whereas, the load to produce a displacement (deflection) of 1 mm in a bending test using the same test piece as the aforementioned one is 2.8 N (280 gf), which is calculated in accordance with Equation (6) below obtained from Equation (4) above:
Thus, it is obvious that the bending test is advantageous in measurement accuracy and load application cost (see Japanese Laid-open Patent Publication No. 2003-232709).
The stress-strain characteristic of a material beyond the point of yield stress, however, cannot be obtained from the displacement-load curve obtained in a three-point bending test.
In the bending test, a large displacement can be produced with a very small load. Moreover, bending deformation of a test piece made of very thin film, which is not suitable for the tensile test, can also be calculated, enabling such a test piece to undergo a material property test. On the other hand, in the tensile test, as described above, the stress-strain relationship can be directly calculated by applying a specific stress (σ=P/A) to a material and calculating the strain ε=δ/L from the material elongation δ=L′−L. The bending test has a problem in that the stress-strain relationship cannot be calculated directly from the relationship between the bending displacement δ and the load P. This is because the stress-strain relationship is nonlinear, making it difficult to estimate the original stress-strain relationship from the displacement-load relationship.
When the stress-strain relationship is linear, Hooke's law of σ=Eε holds in the stress-strain relationship. In addition, the stress and the external load are proportional to each other with a relationship σ=kP (σ=P/A for the tensile test, and σ=M/Z=PL/(4Z)=3PL/(2bh2) for the bending test).
The strain and the displacement are also proportional to each other as expressed by ε=kδ (ε=δ/L for the tensile test, and ε=6δh/L2 for the bending test). Therefore, the stress and the strain can be estimated from the displacement δ and the load P, which are measurable with ease.
Even if the stress-strain relationship is nonlinear, the stress can be estimated from the load as long as there is a proportional relationship expressed by σ=kP between the stress and the load. However, the load and the stress beyond the point of yield of the material are not proportional to each other, and the relationship there between changes nonlinearly, following the stress-strain curve of the material. Specifically, the relationship ∫σ·y·dA=∫σ(y)·y2·dy=M=PL holds.
When a stress over the yield stress acts on a test piece, the internal stress occurring in the test piece changes nonlinearly, following the stress-strain curve, in accordance with a length y in the thickness direction from the neutral axis of the test piece (if the test piece is made of a homogeneous material having a rectangular cross section, the neutral axis lies in the center in the thickness direction). Consequently, the relationship between the maximum stress σ and the external load P also changes nonlinearly.
When the displacement δ is not very large, the strain and the displacement are proportional to each other, the same as in the foregoing case; even if the stress exceeds the yield stress and the stress-strain relationship becomes nonlinear. Specifically, the strain in the bending test is expressed by ε=6δh/L2.
Nevertheless, when the displacement is large, the strain-displacement relationship becomes nonlinear. Therefore, in a test in which the amount of deformation is large and the stress-strain relationship is nonlinear, various nonlinear relationships occur simultaneously, resulting in difficulties in estimating the original stress-strain relationship of the material from the displacement δ and the load P.
In contrast, if the nonlinear stress-strain relationship of a material is known and the shape of a test piece of the material and loading conditions are explicitly provided, it is possible to estimate the load-displacement relationship. For simplicity, a case where the strain-displacement relationship is linear will be described. First, a certain amount of displacement δ is defined, whereby the strain can be calculated from the relationship ε=6δh/L2.
Subsequently, a stress σ corresponding to the strain c is calculated from the stress-strain curve. Lastly, integration of the stress is performed in accordance with the relationship ∫σ(y)·y2·dy=M=PL, whereby a load P to produce the displacement δ is obtained. Thus, the relationship between δ and P can be calculated. Even if the displacement-stress relationship is nonlinear, the same procedure can be taken. First, a certain amount of displacement δ is defined, and the strain ε is obtained by iterative calculation. Subsequently, a stress σ corresponding to the strain ε is calculated from the stress-strain relationship. Lastly, integration of the stress is performed in accordance with the relationship ∫σ(y)·y2·dy=M=PL, whereby a load P to produce the displacement δ is obtained.
As described above, if the stress-strain relationship is known, it is possible to calculate the load-displacement relationship in the three-point bending test. In contrast, even if the load-displacement relationship is known, the stress-strain relationship cannot be calculated. Nevertheless, if a stress-strain relationship close to the genuine stress-strain relationship is reproduced from a few parameters in a certain manner, it is possible to estimate the stress-strain relationship from the load-displacement relationship.